As a conventional damper that uses a fluid having magnetic properties as a working fluid, there is known one having: a cylinder 101; a piston 102; a first fluid chamber 103 and a second fluid chamber 104 partitioned in the cylinder 101 by the piston 102; a magneto-rheological fluid 105 filling the first fluid chamber 103 and the second fluid chamber 104; an electromagnet 108 formed of a coil obtained by winding an electric wire around a groove portion 102a formed on an outer periphery of the piston 102; and an external power feed control device 110 that supplies an electric power to the electromagnet 108 through a wiring line 109, wherein the power feed control device 110 feeds an electric power to the electromagnet 108 to form a magnetic field in a fluid-flow path 107 and a fluid flow resistance of the magneto-rheological fluid 105 passing through the fluid-flow path 107 is increased by a function of this magnetic field to adjust a damping force of the damper as shown in FIG. 18.    Patent Document 1: Japanese Patent Application Laid-open No. 2004-316797
Further, in damping force control of a conventional damping force control type damper that is also called a semi-active damper including a sensor that measures a displacement amount or a speed of a piston and a control device that controls a coil voltage in real time based on measurement data from the sensor, a control method of intensifying a damping force when the damper is effective for suppressing vibrations and weakening the damping force when the damper is not effective for suppressing vibrations is extensively used. Specifically, assuming that one side in a cylinder axial direction is positive and the other side in the same is negative in regard to a relative speed Vr and a relative displacement amount Xr of the piston with respect to the cylinder, there is considered, e.g., a control rule that the damping force is maximized in a case where a direction of relative movement of the piston with respect to the cylinder is different from a direction of displacement from a neutral position when the piston remains in an initial setting state and is in a standby mode without being displaced, i.e., in a case where Vr>0 and Xr<0, or Vr<0 and Xr>0, and also Vr×Xr<0, and that the damping force is minimized in a case where a direction of relative movement of the piston with respect to the cylinder is the same as a direction of displacement from the neutral position, i.e., in a case where Vr>0 and Xr>0, or Vr<0 and Xr<0, and also Vr×Xr>0 (S. Rakheja: Vibration and Shock Isolation Performance of a Semi-Active “On-Off” Damper, American Society of Mechanical Engineers, Journal of Vibration, Acoustics, Stress, and Reliability in Design 107, pp. 398-403, 1985). When the damping force is adjusted by using a relationship between a relative speed and a relative displacement amount of the piston with respect to the cylinder in this manner, the damper can demonstrate an excellent damping effect.